The function $g$ has derivatives of all orders for all real numbers. The Maclaurin series for $g$ is given by $g ( x ) = \sum _ { n = 0 } ^ { \infty } \frac { ( - 1 ) ^ { n } x ^ { n } } { 2 e ^ { n } + 3 }$ on its interval of convergence.
(a) State the conditions necessary to use the integral test to determine convergence of the series $\sum _ { n = 0 } ^ { \infty } \frac { 1 } { e ^ { n } }$. Use the integral test to show that $\sum _ { n = 0 } ^ { \infty } \frac { 1 } { e ^ { n } }$ converges.
(b) Use the limit comparison test with the series $\sum _ { n = 0 } ^ { \infty } \frac { 1 } { e ^ { n } }$ to show that the series $g ( 1 ) = \sum _ { n = 0 } ^ { \infty } \frac { ( - 1 ) ^ { n } } { 2 e ^ { n } + 3 }$ converges absolutely.
(c) Determine the radius of convergence of the Maclaurin series for $g$.
(d) The first two terms of the series $g ( 1 ) = \sum _ { n = 0 } ^ { \infty } \frac { ( - 1 ) ^ { n } } { 2 e ^ { n } + 3 }$ are used to approximate $g ( 1 )$. Use the alternating series error bound to determine an upper bound on the error of the approximation.

Show that the limits $$\lim_{a \rightarrow +\infty} \int_0^a \sin(x^2) \mathrm{d}x \text{ and } \lim_{a \rightarrow +\infty} \int_0^a \cos(x^2) \mathrm{d}x$$ exist and are finite.

Show that the limits $$\lim _ { a \rightarrow + \infty } \int _ { 0 } ^ { a } \sin \left( x ^ { 2 } \right) \mathrm { d } x \quad \text { and } \lim _ { a \rightarrow + \infty } \int _ { 0 } ^ { a } \cos \left( x ^ { 2 } \right) \mathrm { d } x$$ exist and are finite.